Similarity signature curves for forming periodic orbits in the Lorenz system

In this paper, the short-periodic orbits of the Lorenz system are systematically investigated by the aid of the similarity signature curve, and a novel method to find the short-period orbits of the Lorenz system is proposed. We derive the similarity invariants by the equivariant moving frame theory, and then the similarity signature curve occurs along with them. Our analysis shows that the trajectory of the Lorenz system can be described by two completely different states. One is a stable state where the trajectory rotates around an equilibrium point. The other is a mutation state where the trajectory transitions to another equilibrium point. In particular, the similarity signature curve of the Lorenz system presents a more regular behavior than its trajectories. Additionally, with the assistance of the sliding window method, the quasi-periodic orbits can be detected numerically. Furthermore, all periodic orbits with period p <= 8 in the Lorenz system are found, and their period lengths and symbol sequences are calculated.